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Question 13.3: Use Table 13.1 to integrate the following functions: (a) x^4......

Use Table 13.1 to integrate the following functions:

(a) x^4

(b) \cos k x,  where k is a constant

(c)  \sin (3 x+2)

(d) 5.9

(e) \tan (6 t-4)

(f)  \mathrm{e}^{-3 z}

(g)  \frac{1}{x^2}

(h) \cos 100 n \pi t,  where n is a constant

Step-by-Step
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(a) From Table 13.1, we find  \int x^n \mathrm{~d} x=\frac{x^{n+1}}{n+1}+c, n \neq-1 \text {. }  To find  \int x^4 \mathrm{~d} x \text { let } n=4  we obtain

\int x^4 d x=\frac{x^5}{5}+c

(b) From Table 13.1, we find  \int \cos (a x) \mathrm{d} x=\frac{\sin (a x)}{a}+c.  In this case a = k and so

\int \cos k x d x=\frac{\sin k x}{k}+c

Note that a, b, n and c are constants. When integrating trigonometric functions, angles must be in radians.

(c) From Table 13.1, we find  \int \sin (a x+b) \mathrm{d} x=\frac{-\cos (a x+b)}{a}+c.  In this case a = 3 and b = 2, and so

\int \sin (3 x+2) d x=\frac{-\cos (3 x+2)}{3}+c

(d) From Table 13.1, we find that if k is a constant then  \int k \mathrm{~d} x=k x+c .  Hence,

\int 5.9 d x=5.9 x+c

(e) In this example, the independent variable is t but nevertheless from Table 13.1 we can deduce

\int \tan (a t+b) \mathrm{d} t=\frac{\ln |\sec (a t+b)|}{a}+c

Hence with a = 6 and b = −4, we obtain

\int \tan (6 t-4) \mathrm{d} t=\frac{\ln |\sec (6 t-4)|}{6}+c

(f) The independent variable is z but from Table 13.1 we can deduce  \int \mathrm{e}^{a z} \mathrm{~d} z=\frac{\mathrm{e}^{a z}}{a}+c.  Hence, taking a = −3 we obtain

\int \mathrm{e}^{-3 z} \mathrm{~d} z=\frac{\mathrm{e}^{-3 z}}{-3}+c=-\frac{\mathrm{e}^{-3 z}}{3}+c

(g) Since  \frac{1}{x^2}=x^{-2},  we find

\int \frac{1}{x^2} \mathrm{~d} x=\int x^{-2} \mathrm{~d} x=\frac{x^{-1}}{-1}+c=-\frac{1}{x}+c

(h) When integrating  \cos 100 n \pi t  with respect to t, note that  100 n \pi  is a constant. Hence, using part (b) we find

\int \cos 100 n \pi t \mathrm{~d} t=\frac{\sin 100 n \pi t}{100 n \pi}+c

Table 13.1
The integrals of some common functions.
f(x) \int f(x) d x f(x) \int f(x) \mathrm{d} x
k, constant kx + c \cos (a x+b) \frac{\sin (a x+b)}{a}+c
x^n \frac{x^{n+1}}{n+1}+c \quad n \neq-1 \tan x \ln |\sec x|+c
x^{-1}=\frac{1}{x} \ln |x|+c \tan a x \frac{\ln |\sec a x|}{a}+c
\mathrm{e}^x \mathrm{e}^x+c \tan (a x+b) \frac{\ln |\sec (a x+b)|}{a}+c
\mathrm{e}^{-x} -\mathrm{e}^{-x}+c \operatorname{cosec}(a x+b) \frac{1}{a}\{\ln \mid \operatorname{cosec}(a x+b)
\mathrm{e}^{a x} \frac{\mathrm{e}^{a x}}{a}+c -\cot (a x+b) \mid\}+c
\sin x -\cos x+c \sec (a x+b) \frac{1}{a}\{\ln \mid \sec (a x+b)
\sin a x \frac{-\cos a x}{a}+c +\tan (a x+b) \mid\}+c
\sin (a x+b) \frac{-\cos (a x+b)}{a}+c \cot (a x+b) \frac{1}{a}\{\ln |\sin (a x+b)|\}+c
\cos x \sin x+c \frac{1}{\sqrt{a^2-x^2}} \sin ^{-1} \frac{x}{a}+c
\cos a x \frac{\sin a x}{a}+c \frac{1}{a^2+x^2} \frac{1}{a} \tan ^{-1} \frac{x}{a}+c

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